To extend the sketch above to the general case, with nonzero error,
requires some modifications. At the most basic level, we may consider
what happens when we allow for deterministic classical strategies that
succeed with probability less than 1 in the test. Such strategies
may have “mislabeling” error: the points function
\(f\)  may be imagined to be a global codeword of \(\mathcal {C}^{\otimes 3}\)  that has been
corrupted at a small fraction of the points. This type of error is
handled by the analysis in [1]}. The main modification to the zero-error
sketch above is a careful analysis of the probability that the pasting
step produces a “good” interpolated codeword for a randomly chosen
pair of planes \(S_{z_1}, S_{z_2}\) . This analysis makes use of the
distance property of the code together with combinatorial properties of the
point-line test itself (namely, the expansion of the hypercube graph associated with \([n]^m\) ).
